Virginia SOL Algebra I: a complete guide to solving quadratic equations (A.EI.6)

What this category demands

This guide covers solving quadratic equations, standard A.EI.6, part of the Equations and Inequalities reporting category on the Virginia Algebra I SOL. You need four methods and the judgment to pick the right one: factoring, square roots, completing the square, and the quadratic formula, plus the discriminant for counting solutions. Each dot-point page has its own practice: solving quadratics by factoring, solving quadratics by square roots, completing the square, and the quadratic formula and the discriminant.

Solving by factoring

Put the quadratic in standard form $ax^2 + bx + c = 0$, factor, then use the zero product property: set each factor to zero and solve. The solutions are the zeros (the $x$-intercepts of the parabola). Remember the signs flip from the factors: $(x + 5)(x - 3) = 0$ gives $x = -5$ and $x = 3$.

Solving by square roots

When there is a squared term and no linear term, isolate the square and take the square root of both sides with a $\pm$: $x^2 = 49$ gives $x = \pm 7$, and $(x - 3)^2 = 16$ gives $x - 3 = \pm 4$. A square equal to a negative has no real solution.

Completing the square

Add $\left(\frac{b}{2}\right)^2$ to turn $x^2 + bx$ into a perfect square. To solve, move the constant over, add that number to both sides, factor, and square-root. The same move rewrites a quadratic in vertex form $a(x - h)^2 + k$.

The quadratic formula and the discriminant

The quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves any quadratic. Identify $a$, $b$, $c$ with signs, substitute, and simplify. The discriminant $b^2 - 4ac$ counts the real solutions: positive gives two, zero gives one (a double root), negative gives none.

How this category is examined

• Fill-in-the-blank. Solve a quadratic by any method and type both solutions, exact radicals included.
• Multiple choice. Pick the solution set, identify the completing-the-square constant, or use the discriminant to count real solutions.
• Drag-and-drop. Order the steps of a method.

Check your knowledge

Work these as you would for credit on the online test.

1. Solve $x^2 + 2x - 15 = 0$ by factoring. (2 points)
2. Solve $x^2 + 4x = 0$. (1 point)
3. Solve $x^2 = 81$. (1 point)
4. Solve $(x + 2)^2 = 25$. (2 points)
5. What constant completes the square for $x^2 + 10x$? (1 point)
6. Solve $x^2 - 4x - 5 = 0$ by completing the square. (2 points)
7. Solve $x^2 - 5x + 6 = 0$ with the quadratic formula. (2 points)
8. How many real solutions does a quadratic with discriminant $-7$ have? (1 point)
9. Factor and solve $x^2 - 9 = 0$. (1 point)
10. Solve $2x^2 = 18$. (1 point)